1. Introduction

Modern industrial systems are affected usually by various event of faults such as, loss of actuator effectiveness, failures or offsets of actuators/sensors, deviations of output measurement, etc. Indeed, the presence of fault causes an unacceptable performances of design controllers, thus deteriorating the overall system execution, and so leading to wrong dangerous situations.

Thus, it is important to encourage the development of research on fault tolerant control (FTC), which is divided on two types. The first one, the so-called passive FTC, is focused on to conceive a robust controller against disturbances and uncertainties. A key limitation is that the system stability cant be guaranteed in the presence of faults.

Nevertheless, based on online fault estimation (FE), such as the size and the shape, active FTC can develop robust controller such that the fault effects are eliminated and the system stability is achieved. In the literature, several research results on the FTC techniques are documented, see for example [1–9], and the references therein.

In industrial processes, most of systems are described by nonlinear mathematical models. Takagi-Sugeno (TS) fuzzy systems [10] provide a powerful tool to approximate nonlinear characteristics. T-S fuzzy systems are nonlinear models represented by a set of local linear models. By fuzzy blending of linear representations with appropriate membership functions, the overall fuzzy model of the system is achieved, which greatly simplifies the analysis and control for complex nonlinear systems. Therefore, excellent results in FE and FTC problems of T-S fuzzy systems are developed in [11–17]. In [18], a FTC is designed for TS fuzzy systems subject to actuator faults. However, this result must verify the rank condition, which is really hard to fulfill for many practical systems. In [19], the problem of FE and FTC for a T-S fuzzy systems with uncertainties and actuator faults is investigated without the requirement of rank condition. It deals only with constant faults, however, the faults are time-varying in many real systems. In [20], a sliding mode observer (SMO) is designed to estimate sensor fault for nonlinear stochastic systems for FTC. However, the actuator fault is not considered. In [21], a FTC is developed for T-S fuzzy systems affected by actuator faults. Since in many practical systems, actuator and sensor faults may occur at the same time and uncertainties may exist. It is desirable to consider actuator and sensor faults under one unified framework. For example, in [22], a SMO-based FE and FTC is designed for a class of nonlinear systems subject to actuator and sensor faults. A major disadvantage of this approach is the appearance of chattering mode. In [23], FE and FTC problems are studied to simultaneously estimate sensor and actuator faults using two observers and filter. However, this method is more expensive since it requires a high additional computation cost. In [24], a robust adaptive observer is developed to simultaneously estimate state and both sensor and actuator faults for nonlinear systems despite the presence of disturbances. A FTC law is applied to stabilize the closed-loop system and compensate the fault effects. However, sufficient conditions of observer and controller gains are formulated in an unified optimization problem and computed by solving a set of LMIs only in single step. Unfortunately, these results need the knowledge of the upper bounds of faults. If the information of fault is unknown, the SMO cannot be obtained.

The aim of this work is to address fault estimation and fault tolerant control problems for T-S fuzzy systems subject to simultaneously actuator faults, sensor faults and uncertainties. First, a novel robust adaptive SMO is proposed to estimate the states and both actuator and sensor faults using equivalent output error injection approach. Then, based on online fault information a static outputfeedback fault-tolerant control (SOFFTC) is designed to compensate the fault effects and stabilize the closed-loop system. All the design conditions are formulated in an optimization problem under LMIs constraints. Finally, the simulation result of an inverted pendulum with cart system is given to prove the effectiveness of the proposed method.

The main contributions of the present work are the following:

1. A novel fuzzy adaptive SMO is designed for the estimation of states and faults in a T-S fuzzy systems affected by simultaneously actuator faults, sensor faults and uncertainties. Robustness against uncertainties is analyzed using the H_∞ technique to reduce its effect.
2. Most existing SMO design methods such as those reported in [22–24] assume that the value of the upper bounds of actuator faults ρ_a and sensor faults ρ_s is known. If the information of fault is unknown or exceeds the admissible value, these methods cannot be feasible. To overcome this problem, a new adaptive law is constructed to estimate the upper bounds online.
3. The problem of both actuator and sensor FE under one unified framework for T-S fuzzy systems is investigated. Whereas, many researchers have considered only sensor faults [25–27] or actuator faults [28,29].
4. Basedon the FE, a SOFFTC is designed to effectively accommodate the influence of fault and ensure the stability of the resulting closed-loop system. The proposed method is easily be implemented in practice and is much simpler than dynamic output feedback fault tolerant controller.
5. Sufficient conditions of the observer and controller are formulated in an optimization problem under LMIs constraints which can be designed separately.

The rest of this paper is organized as follows: Section 2 presents the problem formulation and preliminaries. The design of the observer and the analysis of the stability of the error dynamics are given in Section 3. FE is studied in Section 4. Section 5 gives the SOFFTC scheme. Finally, simulation example in Section 6 validates the efficiency of the proposed algorithm.

2. Problem Formulation and Preliminaries

Consider a TS fuzzy model with actuator faults, sensor faults and uncertainties. The ith rule of the T-S fuzzy model is of the following form:

Plant Rule i: If ξ₁(t) is µ_1,i and … ξ_g(t) is µ_g,i, Then

where x(t) ∈ Rⁿ represents the state vector; u(t) ∈ R^m is the input; y(t) ∈ R^p is the output; y_c(t) ∈ R^p1 is the controlled output; A_i, B_i, M_i, E_i, C_i, N and C_ci are real known constant matrices with appropriate dimensions; f_a(t) : R⁺ → R^q and f_s(t) : R⁺ → R^h represent additive actuator fault and sensor fault vector, respectively; d(x,u,t) ∈ R^l models the uncertainties, which is assumed to belong to L₂ [0, ∞); the pairs (A_i,C_i) are observable, and the pairs (A_i, B_i) are controllable; ξ_j(j = 1,…,g) are the premise variables, and µ_j,i(j = 1,…,g;i = 1,…,k) are fuzzy sets; g and k are the number of premise variables and IF-THEN rules, respectively. The fuzzy model is given by:

• θ_ij µ_j(t) and here θ_ij(.) stands for the order of the membership

function of θ_ij. It is assumed that

where ρ_a and ρ_s are unknown positive scalars.

Assumption 2 [30]. The actuator fault distribution matrices M_i in (2) satisfy:

Lemma 3 [33]. Under Assumption 2, there exists coordinate trans where A_11,i ∈ Rq×q, A_22,i ∈ R(n−q)×(n−q), B_1,i ∈ Rq×m, M_1,i ∈ Rq×q, E_1,i ∈ Rq×l, N₂ ∈ R(p−q)×h, C_11,i ∈ Rq×q and C_22,i ∈ R(p−q)×(p−q) is invertible, i = 1,…,k.

Through coordinate transformations, the system (2) is converted into the following two subsystems:

where S ₁₁,_i ∈ R^(p−q)×p and S ₂₂,_i ∈ R^q×p. The variable z₁(t) can be obtained by:

Lemma 4 [33]. The pair (A_0,i,C_0,i) is observable, if the pair (A_22,i,C_22,i) is detectable, i = 1,…,k. Then, there exists matrices L_i,

h     i having the special structure L_i =     L_1,i                 0              , such that A_22,i+L_iC_22,i is stable, i = 1,…,k.

Let   the          transformation     of            coordinates           h(t)          = ^h h^T1 (t)               h^T₂ (t) ^iT = T_L,_iz₀(t) with

Therefore, T-S fuzzy subsystems (12) and (13) can be rewritten respectively as:

For system (19), we construct the following adaptive SMO:

where zˆ₁(t), hˆ₁(t) and υˆ₁(t) denote, respectively, the estimated z₁(t), h₁(t) and _,i ∈ R^q×q is a stable matrix and ν_1,i(t) is defined

where P₁ ∈ R^q×q > 0 is the Lyapunov matrix for A₁₁^s _,i, ρˆ_a is adaptive parameter to estimate the unknown parameter ρ_a, and the scalar ρˆ_a is introduced using an update law with constant σ₁ > 0.

For system (20), we design the following adaptive SMO:

where P₀₂ ∈ R^{(p−q)×(p−q)} > 0, ρˆ_s is adaptive parameter to estimate the unknown parameter ρ_s, and the scalar ρˆ_s is introduced using an update law ρ˙ˆs = σ2    N2T P02 (υ3(t) − υˆ3(t))    (24) with constant σ₂ > 0.

Let us define e₁(t) = z₁(t) − zˆ₁(t), e₂(t) = h₁(t) − hˆ₁(t) and e₃(t) = h₂(t) − hˆ₂(t), then the error dynamic system as follows:

with H1 ∈ Rq×q, H2 ∈ R(n−q)×(n−q) and H3 ∈ R(p−q)×(p−q). The adaptive SMO design method under H_∞ performance to be addressed in this work is

• The observer error dynamics system (25) with d(x,u,t) = 0 is asymptotically stable, namely, there is no uncertainty;
• For a given γ₁ > 0. The following H_∞ performance is satisfied:

for all T > 0 and d(x,u,t) ∈ L₂ ^h 0             ^∞  under zero initial conditions.

3.1     Stability analysis

Theorem 1. Consider T-S fuzzy system (2) under Assumptions 13. The observer error dynamics system (25) is asymptotically stable and satisfy (28) with attenuation level γ₁ > 0, if there exist matrices P₁ > 0, P₀₁ > 0, P₀₂ > 0, X_i, Y_i, i = 1,…,k, such that:

Minimize γ₁ subject to

If the optimization problem is solved, then we can obtain the following observer gains

Proof. Let the following Lyapunov functional candidate:

Using the definition of ν_1,i(t) and the bound of f_a(t), we have e^T₁ (t)P₁M_1,i(fa(t) − ν₁,_i(t)) + σ 1₁ ρ˜_a(−ρˆ˙_a)

When d(x,u,t) = 0, we have

If Λ_i < 0, then V˙ (t) < 0, which implies that e → 0 as t → ∞.

Therefore, the error dynamics system is asymptotically stable.

When d(x,u,t) , 0, we define

The previous inequalities are nonlinear because of P₀₁L_i and P₀₂K_i. This problem can be solved using the variable change X_i = P₀₁L_i and Y_i = P₀₂K_i. Applying the Schur complement, we can obtain the LMI form (29). So if J₁(t) < 0, the error dynamics system (25) is stable satisfying the H_∞ performance (28). .

3.2     Sliding motion analysis

For system (25), Let

Theorem 2. Consider system (2) satisfying Assumptions 13. The system error dynamics (25) can be driven to the sliding surface S in finite time and remain on it if the LMI (29) is solvable and the gains

Then the reachability condition [34] is verified.

4. Fault Estimation

From Theorem 2, an ideal sliding mode take place on S and e˙₁(t) = e₁(t) = 0. Consequently, the error dynamics of e₁(t) becomes:

where ν_1eq(t) denotes the equivalent term [34] replaced by:

where β1 =    M1−1A12   maxσmax(H−1) and β2 =    M1−1E1   max. Thus for a small      γ₁β₁ + β₂, f_a(t) can be estimated as:

5. Fault Tolerant Controller Design

Define corrected output as:

where e_fs(t) = f_s(t) − ^fˆ_s(t).

A SOFFTC law [35] is designed as follows:

The gain G_i is designed so that B_iG_i = M_i where B⁺_i is the pseudo inverse of B_i [36]. It follows that

^h e^Tf s(t) e^T_fa(t) d^T(x,u,t) ⁱ . The control purpose in this paper for the closed-loop fuzzy system (63) is to design a SOFFTC (60) such that

• The closed-loop fuzzy system (63) with (ϕ(t) = 0) is asymptotically stable .
• For a given scalar γ₂ > 0, the following H_∞ performance is satisfied:

for all L > 0 and ϕ(t) ∈ L₂ [0, ∞) under zero initial conditions.

Theorem 3. The closed-loop system (63) is asymptotically stable and satisfy the H_∞ performance index (64), if there exist matrix P¯_x > 0, matrices R, S¯ _i and scalar  > 0, such that:

where

with Υij = (Ai + BiK¯ jCi)T Px + Px(Ai + BiK¯ jCi) + CiTCj. Thus, J_x(t) < 0, if

where Υ¯ _ij = (A_i + B_iK¯ _jC_i)^T P_x + P_x(A_i + B_iK¯ _jC_i). Premultiplying and postmultiplying by Π = diagnP⁻_x¹, P⁻_x¹, I, I, I^o

Notice that the matrix inequality Ω_ij < 0 is a Bilinear Matrix Inequalities (BMIs). Denoting C_iP¯_x = RC_i and K¯ _jR = S¯ _j, so that

with

If (84) is verified, we have J_x(t) < 0, which can guarantee the closed-loop system is asymptotically stable.

6. Simulation Example

In this example, The inverted pendulum with cart system [37] is considered:

where x₁(t) and x₂(t) are the angular position and velocity, respectively; x₃(t) and x₄(t) are the cart position and velocity, respectively; m is the pendulum mass, M is the cart mass, g is the gravity constant, and a = 1/(m + M) . The values of the parameters used in this simulation are m = 0.2kg, M = 0.8kg, l = 0.5m, and L = 2m .

Then, we get the TS fuzzy system with the following fuzzy rules:

Rule1: If x₁(t) is about 0, Then

( x˙(t) = A₁x(t) + B₁(u(t) + f_a(t)) + E₁d(t) y(t) = C₁x(t) + N f_s(t)

Rule1: If x₁(t) is about ± , Then

we set d(x,u,t) = − f_c + m l x₂² (t) sin(x₁ (t)) where f_c = ρ sign(x₄ (t)) with ρ = 0.05 . The membership functions for rules 1 and 2 are chosen based on the method of sector nonlinearity [38] as follows:

By Choosing H₁ = 1, H₂ = I₃ and H₃ = I₂, we can solve the optimization problem of Theorem 1 using Matlab LMI Toolbox and we obtain γ₁ = 0.3614, A₁₁^s _,1 = −3.2457, A₁₁^s _,2 = −3.2457,

Solving the optimization problem in Theorem 3 results the following controller gain matrices for a minimum attenuation level γ₂ = 1.6128:

                  K¯₁ = ^h −2.7838     −6.8067      −11.1030 ⁱ

                   K¯₂ = ^h −2.1520     −7.5055      −11.8265 ⁱ

We simulate the closed-loop system be choosing σ₁ = 10, σ₂ = 15, δ₁ = 0.01, δ₂ = 0.02, l_a,1 = l_a,2 = 10 and l_s,1 = l_s,2 = 12, initial conditions x₁₀ = π/20, x₂₀ = 0, x₃₀ = 2 and x₄₀ = 0. f_a(t) and f_s(t) are assumed that, respectively

The simulation results of this example are given in Figures 1-5. Figures 1 and 2 show that the proposed adaptive SMO can estimate the actuator and sensor faults simultaneously despite the presence of uncertainties.

Actuator fault estimation

Figure 1: The actuator fault f_a(t) and its estimation.

Figure 2: The sensor fault f_s(t) and its estimation.

Figure 3: Output y₁(t) under the static output feedback FTC.

Figure 4: Output y₂(t) under the static output feedback FTC.

Figure 5: Output y₃(t) under the static output feedback FTC.

Simulation results for the systems outputs response are illustrated in Figures 3-5. It is clear to see that the outputs without SOFFTC do not converge to the outputs of the fault-free model (i.e. without any fault). However, the outputs’s trajectories of the system 7 with SOFFTC reach the outputs of nominal model.

Conclusion

In this paper, we have considered the problems of FE and FTC for T-S fuzzy systems with uncertainties, actuator and sensor faults, simultaneously. Using the H_∞ optimization technique, an adaptive fuzzy sliding mode observer has been firstly designed to estimate the system state, actuator and sensor faults, simultaneously. Secondly, using the information of online fault estimates, a novel SOFFTC has been developed to compensate the faults and stabilize the closedloop system. Thus, sufficient condition for the existence of the proposed ASMO and SOFFTC has been formulated in terms of LMIs. Finally, a simulation example was used to show the effectiveness of the proposed methods.

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